Consider the symbiotic relationship between a cleaner sh and the host that they clean. In each interaction
Question:
Consider the symbiotic relationship between a cleaner fish and the host that they clean. In each interaction a cleaner fish (row player) can cooperate (C) by properly cleaning its host or defect (D) by taking little bites out of its host's mucosal membrane while cleaning its mouth. The host fish (column player) can also cooperate (C) by holding still and keeping its mouth wide open while it is being cleaned, or defect (D) by trying to close its mouth on the cleaner fish and eat it. The interaction is summarized in the payoff matrix (contributions to their fitness) below. α and the β indicate payoffs that are missing. A) Based on observations you know that, on average, cleaner fish cooperate in 2/3 of interactionswith host fish, while host fish cooperate in 1/2 of the interactions. You hypothesize that these probabilities represent a mixed strategy Nash equilibrium. Determine α, β. B) What is the expected utility in an encounter for cleaner fish? And for host fish? We know that x = [2/3, 1/3] = (0.66, 0.33) y = [½, ½] = (0.5, 0.5). But how do you define α, β? Likewise what utility is to be expected? Suppose cleaner fish, in their observations of host fish, figured out that the host fish cooperate with probability 1/2 (the same proportion you observed). Could this information be exploited by the cleaner fish? What is the most preferable outcome for both fish? Can they achieve it?.
